Augmenting the Connectivity of Planar and Geometric Graphs

In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to find a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge(or vertex-) disjoint s–t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution. Submitted: July 2010 Reviewed: August 2012 Revised: August 2012 Accepted: August 2012 Final: August 2012 Published: September 2012 Article type: Regular Paper Communicated by: M. T. Goodrich Supported by grant WO 758/4-3 of the German Science Foundation (DFG). E-mail address: [email protected] (Ignaz Rutter) 600 I. Rutter, A. Wolff Augmentation of Planar and Geometric Graphs